Thurston's Geometrization theorem implies that an acylindrical
hyperbolic manifold M admits a unique hyperbolic metric whose convex
core has totally geodesic boundary. We show that this is the most
efficient hyperbolic metric on M, in the sense that it is the hyperbolic
metric whose convex core has least possible volume.

The result above follows from an extension of work of Besson, Courtois
and Gallot into the setting of Alexandrov spaces with lower bounds on
curvature. This extension also has implications for volumes of hyperbolic
cone manifolds.